Photoionization or Bound Free Transitions

A bound electron in an atom is ejected by an incident photon and becomes a free electron (Fig. 8.1). Such a bound-free (bf) absorption process occurs only for photon energies higher than the ionization potential of the considered level n, where R^ = 2n2mee4/(ch3) = 1.09737 x 105 cm-1 is the Rydberg constant. This expression also defines an absorption edge, i.e., the lowest frequency v* for bound-free absorption,

bound - free free - tree

Fig. 8.1 Schematic illustration of the bound-free and free-free absorptions bound - free free - tree

Fig. 8.1 Schematic illustration of the bound-free and free-free absorptions

The approximation of an hydrogenic atom has been made above, which is (partly) justified by the fact that elements are highly ionized in the stellar interiors. The absorption coefficient of a photon of frequency V by an electron on a level n of an ion of charge Zj is

The factor gbf is a dimensionless factor, called the Gaunt factor for bf absorption. It corrects the above semi-classical expression for quantum effects, in general gbf — 1 and varies slowly with n and v. When an atom has several bound electrons, the charge Zj must be replaced by an effective charge, accounting for the screening effects of bound electrons (Sect. 9.4). We note that heavy elements (with high Zj) are important absorbers and that absorption is stronger at longer wavelengths. At the edge frequency v*, the absorption goes like

1 n6

with (8.18). The absorption from the highly excited states is higher, contrarily to the first impression from (8.19). We thus have

This means that the absorption coefficient for a given element at the edge frequency scales like T-1/2. For an element of relative abundance Xj in mass fraction, the corresponding opacity is

/ s / ,ne( j, n)Xj Kbf(v, j, n) = abf(v, j, n) ^, ' j , (8.22)

Aj mu where ne( j, n) is the average number of bound electrons in the n shell of element j. It is estimated with the Saha-Boltzmann equation (7.23), assuming that the medium is highly ionized so that the ions will have at most one bound electron.

As frequency increases, the opacity coefficient decreases with v-3 (8.19), but suddenly the energy is high enough to eject electrons from a deeper shell and the coefficient raises steeply to decline again in v-3 until the next jump when the energy is high enough to eject an electron from a deeper shell. Then, the summation has to be made on the various elements j:

Electron degeneracy, if present, decreases Kbf (v), since many cells of the phase space of free electrons are occupied. Before taking the Rosseland average (3.22), one must sum up all the various absorption processes. For specific purpose, one may take the mean Rosseland opacity for the bound-free transitions only. Let us examine how it scales with T. A dependence of the form k ~ va in the mean Rosseland opacity (3.22) gives a dependence Ta [285], as obtained by integration of (3.22). Thus, the term v-3 in (8.19) makes a dependence like T-3. With account that the edge frequency scales like T-1/2 (8.21), this gives a general dependence in T-3 5. The mean opacity coefficient behaves roughly like [523], see also [610],

Kbf ~ K0jbf q T-3 5 with K0jbf ~ 4.3 x 1025Z(1 + X) , (8.24)

with q in g cm-3, T in K and Kbf in cm2 g-1. This is the Kramers law for bf transitions. We notice the dependence in the heavy element content Z, which makes Kbf a major effect in stars of solar composition. It applies essentially for T above several 104 K depending on density (see Fig. 8.2).

Some factors of the order of unity are omitted, in particular the Gaunt factor, which is close to 1, and a so-called guillotine factor which accounts for the overestimate of the number of bound electrons given by Saha-Boltzmann relation at low T. Analytical expressions are no longer used for model calculations, but only for simple analytical scaling or estimates. Figure 8.2 shows the total opacity curves. On the right of the peak, we see the decrease approximately predicted by Kramers law for bound-free opacities.

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